Kinematics - PowerPoint Presentation

Slide1

Kinematics

u

sing slides from D.

LuSlide2

Goals of this class

Introduce Forward

K

inematics of

mobile

robots

How Inverse

K

inematics

for

static

and mobile robots can be

derived

Concept of

Holonomy

I

ntuition

on the relationship between inverse

kinematics

and path-planning.Slide3

Kinematics

Kinematics

is the branch of

classical mechanics

which describes the

motion

of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion.

[1]

[2]

[3]

The term is the English version of

A.M. Ampère

's

cinématique,

[4]

which he constructed from the

Greek

κίνημα

kinema

"movement, motion", derived from κινεῖν

kinein

"to move".

[5]

[6

]

Kinematics is

often referred to as the "geometry of

motion.“ Kinematics

begins with a description of the geometry of the system and the initial conditions of

position

, velocity and or

acceleration,

then from geometrical arguments it can determine the position, the velocity and the acceleration of any part of the system.  

In contrast to

Dynamics

, is concerned with relationship between motion of bodies and its causes, the forces

http://en.wikipedia.org/wiki/KinematicsSlide4

Forward/Inverse Kinematics

Forward kinematics

:

f(p, a) = p'

Given pose p and action a,

what is the resulting pose p'?

Inverse kinematics

:

f(p, p') = a

Given poses p and p',

what action a will move from p to p'?Slide5

Forward kinematics of a simple arm

y

2

= sin(

 Slide6

Transformation from end-effector to base

Remember

With cos

α

ß

denoting cos(

α

+ ß) and sin

α

ß

denoting

sin(α+ ß) Slide7

Ho

l

o

n

o

m

i

c

vs.

Non

-­‐Holo

nomic Systems

A system is non-holonomic when closed trajectories in its configuration space may not have it return to its original state.A simple arm is holonomic, as each joint position corresponds to a unique position in space. A train is holonomic.A car and a differential-wheel robot are non-holonomic vehicles.Getting the robot to its initial position requires not onlyto rewind both wheels by the same amount, but also gettingtheir relative speeds right. The speed of each wheel as a function of time matters.The robot's kinematic is holonomic if closed trajectories in configuration space result in closed trajectories in the workspace.Slide8

Ho

l

o

n

o

m

i

c

vs.

Non-­‐Hol

onomic

ManipulatorDiff. WheelsConfiguration Space(set of angles each actuator can be set to) Workspace (the physical space the robot can move to )Slide9

Modeling Wheeled Robots

All motion models are idealized.

No wheel slippage

No axle flex

Wheels don't compress, etc.

Pose - Variables needed for state of robot

Action - Commands to send to robotSlide10

Simple Robot

Aligned to x-axis

One active wheel &

One passive point of contact

Pose? Action?Slide11

Robot Velocity

Action

: Apply velocity

v

for

t

seconds

v in meters per second!Slide12

Wheel Angular Velocity

Action

: Apply angular velocity ѡ for

t

seconds

Wheel has radius

r

ѡ

0

in rotations per second!Slide13

Inverse Kinematic Models

Δx = tv

v = Δx/t

v = (x

2

-x

1

)/t

f(x

1

,x

2

,t) = (x2-x1)/tΔx = 2𝜋rѡ0tѡ0 = Δx/(2𝜋rt)ѡ0 = (x2-x1)/(2𝜋rt)f(x1,x2,t) = (x2-x1)/(2𝜋rt)Slide14

Simple Robot Observations

Models depend on action definition.

Solutions for forward model and inverse model are unique

for this robot

.Slide15

Simple Robot in 2D

Same Robot, at angle θ

Pose?

p = (x,y)

Action?

vSlide16

2D KinematicsSlide17

2D Kinematics

x + tv cos(θ) = x'

y + tv sin(θ) = y'

v = (x'-x)/(t cos(θ))

v = (y'-y)/(t sin(θ))

Observation:

Inverse model has no solution in some parts of spaceSlide18

Differential Drive

Two active wheels (L & R)

Some passive supporting wheels

Pose?

p = (x,y,θ) (taken at center of axis)

Action?

a = (v

L

,v

R

)

v in meters per second!Slide19

v

L

= +k, v

R

=0Slide20

v

L

= +k, v

R

=+k'Slide21

v

L

= +k, v

R

=+kSlide22

v

L

= +k, v

R

=-kSlide23

Wheels Go in Circles

Axle Length

d

b

Wheels travel on circles of circumference

C

L

=2

𝜋(b+d)

C

R

=2𝜋b

ICC

Instantaneous Center of CoordinatesSlide24

Wheels Go In Circles 2

Wheels have same angular velocity around axis of rotation

D

L

D

RSlide25

DD Kinematics

Traveled θ radians around circle

D

L

= C

L

(θ/(2𝜋) )

= 2𝜋(b+d)θ/(2𝜋)

θ = D

L

/(b+d)

D

R = CR (θ/(2𝜋) ) = 2𝜋bθ/(2𝜋)θ = DR/bb = DR/θθ = (DL-DR)/dѡ = θ/t = (DL-DR)/(d t) = (vL-v

R)/d

D

L

D

R

θ

Change of angle

v

R

v

L

v

ѡSlide26

Observations about DD

ѡ

= (v

L

-v

R

)/d

Straight line v

L

=v

R

--> ѡ=0

Smaller d --> larger ѡ for constant |vL-vR|v = (vL+vR)/2Slide27

Forward Kinematics DD

D

L

D

R

θ

x

I

y

I

x

y

In general, integrals cannot be solved analytically

ω

(t) and v(t) are functions of time Slide28

Forward Kinematics

Assume constant v, θ

0

=0Slide29

Inverse Kinematics

Solution with constant velocities does not always exist!Slide30

Using the notation of the books

(

Siegwart

et al,

Correll

)

We have a pose p(x,y,

θ

) and we are interested in their changes

We control

v

L

and vR or in terms of angles on the wheels φL, φR rφSlide31

Transformation of the coordinatesSlide32

In

ver

s

e

Kinematics

o

f

M

o

bile

RobotsSlide33

Additional Assumption

Perfect Instantaneous Activation

No inertia, no massSlide34

Summary

For calculating the forward kinematics of a robot, it is easiest to establish a local coordinate frame on the robot and determine the transformation into the world coordinate first.

Forward and Inverse Kinematics of a mobile robot are performed with respect to the 

speed

 of the robot and not its 

position.

For calculating the effect of each wheel on the speed of the robot, you need to consider the contribution of each wheel independently.

Calculating the inverse kinematics analytically becomes quickly infeasible. You can then plan in configuration space of the robot using path-planning techniques.